Dake L.P. Fundamentals of reservoir engineering

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SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 20

1−y

in the denominators of the second and first terms, respectively, instead of (1−y)

and (1−y)

as in equs. (1.21) and (1.23) of this text.)

Takacs

has determined that the average difference between the Standing-Katz

correlation chart and the analytical Hall-Yarborough method is − 0.158% and the

average absolute difference 0.518%. Figure 1.8 shows an isothermal Z−factor versus

pressure relationship, obtained using the Hall-Yarborough method, for a gas with

gravity 0.85 and at a reservoir temperature of 200°F. The plot coincides, within pencil

thickness, with the similar relation obtained by the application of the method described

in b), above.

The plot shows that there is a significant deviation from the ideal gas behaviour which

is particularly noticeable in the intermediate pressure range at about 2500 psia. At this

pressure, use of the ideal gas equation, (1.13), would produce an error of almost 25%

in calculated gas volumes.

1.6 APPLICATION OF THE REAL GAS EQUATION OF STATE

The determination of the Z−factor as a function of pressure and temperature facilitates

the use of the simple equation

pV ZnRT= (1.15)

to fully define the state of a real gas. This equation is a PVT relationship and in

reservoir engineering, in general, the main use of such functions is to relate surface to

reservoir volumes of hydrocarbons. For a real gas, in particular, this relation is

expressed by the gas expansion factor E, where

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 21

Pressure

(

psia

)

- Facto

0.75

0.85

0.95

1000 2000 3000 4000 5000

Fig. 1.8 Isothermal Z−

−−

−factor as a function of pressure (gas gravity = 0.85;

temperature = 200° F)

volume of n moles of gas at standard conditions

V volume of n moles of gas at reservoir conditions

and applying equ. (1.15) at both standard and reservoir conditions this becomes

sc sc sc

VTZ

VpTZ

(1.24)

For the field units defined in connection with equ. (1.13), and for standard conditions of

= 14.7 psia, T

= (460+60) = 520°R and Z

= 1, equ. (1.24) can be reduced to

E 35.37 (vol / vol)

(1.25)

At a pressure of 2000 psia and reservoir temperature of 180°F the gas whose

composition is detailed in table 1.1 has a Z−factor of 0.865, as already determined in

sec. 1.5(b). Therefore, the corresponding gas expansion factor is

35.37 2000

E 127.8 (vol / vol)

0.865 640

In particular, the gas initially in place (GIIP) in a reservoir can be calculated using an

equation which is similar to equ. (1.2) for oil, that is

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 22

wc i

GV(1S)E

=− (1.26)

in which E

is evaluated at the initial pressure.

Other important parameters which can be conveniently expressed using the equation

of state are, the real gas density, gravity and isothermal compressibility.

Since the mass of n moles of gas is nM, where M is the molecular weight, then the

density is

nM nM Mp

VZnRT/pZRT

== = (1.27)

Comparing the density of a gas, at any pressure and temperature, to the density of air

at the same conditions gives

gas gas

air air

(M / Z)

and, in particular, at standard conditions

gas gas

air air

M28.97

== = (1.28)

where γ

is the gas gravity relative to air at standard conditions and is conventionally

expressed as, for instance, γ

= 0.8 (air = 1).

Therefore, if the gas gravity is known, M can be calculated using equ. (1.28) and

substituted in equ. (1.27) to give the density at any pressure and temperature.

Alternatively, if the gas composition is known M can be calculated as

MnM=

(1.29)

and again substituted in equ. (1.27). The molecular weights of the individual gas

components, M

, are listed in table 1.1. It is also useful to remember the density of air at

standard conditions (in whichever set of units the reader employs). For the stated units

this figure is

(

air

)

= 0.0763 ib/cu.ft

which permits the gas density at standard conditions to be evaluated as

0.0763 g (lbs / cu.ft)

ργ

= (1.30)

The final application of the equation of state is to derive an expression for the

isothermal compressibility of a real gas. Solving equ. (1.15) for V gives

ZnRT

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 23

the derivative of which, with respect to pressure, is

V ZnRT 1 1 Z

pppZp

æö

∂∂

=− −

ç÷

∂∂

èø

and substituting these two expressions in the isothermal compressibility definition,

equ. (1.11), gives

1V 1 1Z

Vp p Zp

∂∂

=− = −

∂∂

(1.31)

In fig. 1.9, a plot of the gas compressibility defined by equ. (1.31) is compared to the

approximate expression.

(1.32)

for the 0.85 gravity gas whose isothermal Z−factor is plotted in fig. 1.8 at 200°F. As can

be seen, the approximation, equ. (1.32), is valid in the intermediate pressure range

between 2000−2750 psia where ∂Z/∂p is small but is less acceptable at very high or

low pressures.

EXERCISE 1.1 GAS PRESSURE GRADIENT IN THE RESERVOIR

1) Calculate the density of the gas, at standard conditions, whose composition is

listed in table 1.1.

2) what is the gas pressure gradient in the reservoir at 2000 psia and 180° F

(Z = 0.865).

EXERCISE 1.1 SOLUTION

1) The molecular weight of the gas can be calculated as

MnM19.91==

and therefore, using equ. (1.28) the gravity is

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 24

Pressure (psia)

GAS

COMPRESSIBILITY

psi

-1

16 ×10

-4

1000 2000 3000 4000 5000

δZ

δp

Fig. 1.9 Isothermal gas compressibility as a function of pressure (gas gravity = 0.85;

temperature = 200° F)

air

M 19.91

0.687 (air 1)

M28.97

== = =

The density at standard conditions can be evaluated using equ. (1.27) as

sc sc

19.91 14.7

0.0524 Ib / cu.ft

Z RT 1 10.732 520

== =

××

or, alternatively, using equ. (1.30) as

sc g

0.0763 0.0524 lb / cu.ft

ργ

2) The density of the gas in the reservoir can be directly calculated using

equ. (1.27), or else by considering the mass conservation of a given quantity of

gas as

(

= (

res

which, using equ. (1.25), can be evaluated at 2000 psia and 180

F as

res

35.37 p

35.37 0.0524 2000

6.696 Ib / cu.ft

ZT 0.865 (180 460)

××

== =

×+

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 25

To convert this number to a pressure gradient in psi/ft requires the following

manipulation.

gas

dp Ib 1 ft 6.696

6.696 0.0465 psi / ft.

dD ft 144ft inch

æö

=××==

ç÷

èø

1.7 GAS MATERIAL BALANCE: RECOVERY FACTOR

The material balance equation, for any hydrocarbon system, is simply a volume

balance which equates the total production to the difference between the initial volume

of hydrocarbons in the reservoir and the current volume. In gas reservoir engineering

the equation is very simple and will now be considered for the separate cases in which

there is no water influx into the reservoir and also when there is a significant degree of

influx.

a) Volumetric depletion reservoirs

The term volumetric depletion, or simply depletion, applied to the performance of a

reservoir means that as the pressure declines, due to production, there is an

insignificant amount of water influx into the reservoir from the adjoining aquifer. This, in

turn, implies that the aquifer must be small (refer sec. 1.4). Thus the reservoir volume

occupied by hydrocarbons (HCPV) will not decrease during depletion. An expression

for the hydrocarbon pore volume can be obtained from equ. (1.26) as

HCPV = V

(1−S

) = G/E

where G is the initial gas in place expressed at standard conditions. The material

balance, also expressed at standard conditions, for a given volume of production G

and consequent drop in the average reservoir pressure ∆p = p

−p is then,

Production GIIP Unproduced Gas

(sc) (sc) (sc)

G G (HCPV)E

GGE

=−

(1.33)

which can be expressed as

=−

(1.34)

or, using equ. (1.25), as

ZZ G

æö

=−

ç÷

èø

(1.35)

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 26

The ratio G

/G is the fractional gas recovery at any stage during depletion and, if the

gas expansion factor E, in equ. (1.34), is evaluated at the proposed abandonment

pressure then the corresponding value of G

/G is the gas recovery factor.

Before describing how the material balance equation is used in practice, it is worthwhile

reconsidering the balance expressed by equ. (1.33) more thoroughly. Implicit in the

equation is the assumption that because the water influx is negligible then the

hydrocarbon pore volume remains constant during depletion. This, however, neglects

two physical phenomena which are related to the pressure decline. Firstly, the connate

water in the reservoir will expand and secondly, as the gas (fluid) pressure declines,

the grain pressure increases in accordance with equ. (1.4).

As a result of the latter, the rock particles will pack closer together and there will be a

reduction in the pore volume. These two effects can be combined to give the total

change in the hydrocarbon pore volume as

d(HCPV) = − dV

+dV

(1.36)

where V

and V

represent the initial connate water volume and pore volume (PV),

respectively. The negative sign is necessary since an expansion of the connate water

leads to a reduction in the HCPV. These volume changes can be expressed, using

equ. (1.11), in terms of the water and pore compressibilities, where the latter is defined

V(GP)

∂

=−

∂

(1.4)

where GP is the grain pressure which is related to the fluid pressure by

d(FP) = − d(GP)

therefore

V(FP) V p

∂∂

=− =

∂∂

(1.37)

where p is the fluid pressure. Equation (1.36) can now be expressed as

d(HCPV) = c

dp + c

or, as a reduction in hydrocarbon pore volume as

d(HCPV) = − (c

+ c

) ∆p (1.38)

where ∆p = p

− p, the drop in fluid (gas) pressure. Finally, expressing the pore and

connate water volumes as

wc i wc

HCPV G

VPV

(1 S ) E (1 S )

== =

−−

and

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 27

wwc

iwc

VPVS

E(1 S )

=×=

−

the reduction in hydrocarbon pore volume, equ. (1.38), can be included in equ. (1.33),

to give

wwc f

wc i

(c S c ) p

G1SE

+∆

æö

=− −

ç÷

−

èø

(1.39)

as the modified material balance. Inserting the typical values of c

= 3 × 10

−6

/psi,

= 10 × 10

−6

/psi and S

= 0.2 in this equation, and considering a large pressure drop

of ∆p = 1000 psi; the term in parenthesis becomes

(3 .2 10)

1101010.013

0.8

−

×+

−××=−

That is, the inclusion of the term accounting for the reduction in the hydrocarbon pore

volume, due to the connate water expansion and pore volume reduction, only alters the

material balance by 1.3% and is therefore frequently neglected. The reason for its

omission is because the water and pore compressibilities are usually, although not

always, insignificant in comparison to the gas compressibility, the latter being defined in

sec. 1.6 as approximately the reciprocal of the pressure. As described in Chapter 3,

sec. 8, however, pore compressibility can sometimes be very large in shallow

unconsolidated reservoirs and values in excess of 100 × 10

−6

/ psi have been

measured, for instance, in the Bolivar Coast fields in Venezuela. In such reservoirs it

would be inadmissible to omit the pore compressibility from the gas material balance.

In a reservoir which contains only liquid oil, with no free gas, allowance for the connate

water and pore compressibility effects must be included in the material balance since

these compressibilities have the same order of magnitude as the liquid oil itself (refer

Chapter 3, sec. 5).

In the majority of cases the material balance for a depletion type gas reservoir can

adequately be described using equ. (1.35). This equation indicates that there is a linear

relationship between p/Z and the fractional recovery G

/G, or the cumulative production

, as shown in fig. 1.10(a) and (b), respectively. These diagrams illustrate one of the

basic techniques in reservoir engineering which is, to try to reduce any equation, no

matter how complex, to the equation of a straight line; for the simple reason that linear

functions can be readily extrapolated, whereas non-linear functions, in general, cannot.

Thus a plot of p versus G

/G or G

, would have less utility than the representations

shown in fig. 1.10, since both would be non-linear.

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 28

G / G

RF G

( a )

(RF) comp

( b )

Fig. 1.10 Graphical representations of the material balance for a volumetric depletion

gas reservoir; equ. (1.35)

Figure 1.10(a) shows how the recovery factor (RF) can be determined by entering the

ordinate at the value of (p/Z)

corresponding to the abandonment pressure. This

pressure is dictated largely by the nature of the gas contract, which usually specifies

that gas should be sold at some constant rate and constant surface pressure, the latter

being the pressure at the delivery point, the gas pipeline. Once the pressure in the

reservoir has fallen to the level at which it is less than the sum of the pressure drops

required to transport the gas from the reservoir to the pipeline, then the plateau

production rate can no longer be maintained. These pressure drops include the

pressure drawdown in each well, which is the difference between the average reservoir

and bottom hole flowing pressures, causing the gas flow into the wellbore; the pressure

drop required for the vertical flow to the surface, and the pressure drop in the gas

processing and transportation to the delivery point. As a result, gas reservoirs are

frequently abandoned at quite high pressures. Recovery can be increased, however,

by producing the gas at much lower pressures and compressing it at the surface to

give the recovery (RF)

comp

, as shown in fig. 1.10(a). In this case the capital cost of the

compressors plus their operating costs must be compensated by the increased gas

recovery.

Figure 1.10(b) also illustrates the important techniques in reservoir engineering,

namely, "history matching" and "prediction". The circled points in the diagram, joined by

the solid line, represent the observed reservoir history. That is, for recorded values of

the cumulative gas production, pressures have been measured in the producing wells

and an average reservoir pressure determined, as described in detail in Chapters 7

and 8.

Since the plotted values of p/Z versus G

form a straight line, the engineer may be

inclined to think that the reservoir is a depletion type and proceed to extrapolate the

linear trend to predict the future performance. The prediction, in this case, would be

how the pressure declines as a function of production and, if the market rate is fixed, of

time. In particular, extrapolation to the abscissa would give the value of the GIIP which

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 29

can be checked against the volumetric estimate obtained as described in secs. 1.2 and

1.3. This technique of matching the observed production pressure history by building a

suitable mathematical model, albeit in this case a very simple one, equ. (1.35), and

using the model to predict future performance is one which is fundamental to the

subject of Reservoir Engineering.

b) Water drive reservoirs

If the reduction in reservoir pressure leads to an expansion of the adjacent aquifer

water, and consequent influx into the reservoir, the material balance equation must

then be modified as

Production GIIP Unproduced Gas

(sc) (sc) (sc)

GG WE

=−

æö

=−−

ç÷

èø

(1.40)

where, in this case, the hydrocarbon pore volume at the lower pressure is reduced by

the amount W

, which is the cumulative amount of water influx resulting from the

pressure drop. The equation assumes that there is no difference between surface and

reservoir volumes of water and again neglects the effects of connate water expansion

and pore volume reduction.

If some of the water influx has been produced it can be accounted for by subtracting

this volume, W

, from the influx, W

, on the right hand side of the equation. With some

slight algebraic manipulation equ. (1.40) can be expressed as

ZZ G G

æö

=− −

ç÷

èø

(1.41)

where W

/G represents the fraction of the initial hydrocarbon pore volume flooded

by water and is, therefore, always less than unity. When compared to the depletion

material balance, equ. (1.35), it can be seen that the effect of the water influx is to

maintain the reservoir pressure at a higher level for a given cumulative gas production.

In addition, equ. (1.41) is non-linear, unlike equ. (1.35), which complicates both history

matching and prediction. Typical plots of this equation, for different aquifer strengths,

are shown in fig. 1.11.

During the history matching phase, a separate part of the mathematical model must be

designed to calculate the cumulative water influx corresponding to a given total

pressure drop in the reservoir; this part of the history match being described as "aquifer

fitting". For an aquifer whose dimensions are of the same order of magnitude as the

reservoir itself the following simple model can be used